Use Properties of Tangents - Ms. Young · Use Properties of Tangents Georgia Goal pUse properties of a tangent to a circle. Performance Standard(s) MM2G3a, MM2G3d Your Notes VOCABULARY

192 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.

Use Properties of TangentsGoal p Use properties of a tangent to a circle.Georgia

PerformanceStandard(s)

MM2G3a, MM2G3d

Your Notes

VOCABULARY

Circle

Center

Radius

Chord

Diameter

Secant

Tangent

6.1

Tell whether the line, ray, or segment A

F

C

E B D

is best described as a radius, chord,diameter, secant, or tangent of (C.

a. }BC b. @##$EA c. ###$DE

Solution

a. }BC is a because C is the center and B is a point on the circle.

b. @##$EA is a because it is a line that intersects the circle in two points.

c. ###$DE is a ray because it is contained in a line that intersects the circle in exactly one point.

Example 1 Identify special segments and lines

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ga2_ntg_06.indd 192ga2_ntg_06.indd 192 4/19/07 3:45:19 PM4/19/07 3:45:19 PM

Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 193

Your Notes

Use the diagram to find the

x

y

1

1

A B

given lengths.

a. Radius of (A

b. Diameter of (A

c. Radius of (B

d. Diameter of (B

Solution

a. The radius of (A is units.

b. The diameter of (A is units.

c. The radius of (B is units.

d. The diameter of (B is units.

Example 2 Find lengths in circles in a coordinate plane

1. In Example 1, tell whether }AB is best described as a radius, chord, diameter, secant, or tangent. Explain.

2. Use the diagram to find (a) the radius of (C and (b) the diameter of (D.

x

y

1

1

D

C

Checkpoint Complete the following exercises.

ga2nt-06.indd 118 4/16/07 9:01:26 AM



Your Notes

THEOREM 6.1

In a plane, a line is tangent to a circle if

O

Pand only if the line is to a radius of the circle at its endpoint on the circle.

Tell how many common tangents the circles have and draw them.

a. b. c.

Solution

a. common b. common c. commontangent(s) tangent(s) tangent(s)

Example 3 Draw common tangents

3. 4.

Checkpoint Tell how many common tangents the circles have and draw them.

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Your Notes

In the diagram, B is a point of B

A Cr49

77 rtangency. Find the radius r of (C.

SolutionYou know from Theorem 6.1 that }AB ⊥ }BC, so nABC is a

. You can use the Pythagorean Theorem.

AC2 5 BC2 1 AB2 Pythagorean Theorem

(r 1 49)2 5 r2 1 772 Substitute.

r2 1 r 1 5 r2 1 Multiply.

r 5 Subtract from each side.

r 5 Divide each side by .

The radius of (C is .

Example 5 Find the radius of a circle

5.

S

TR

5

8

12

6. S

T

R 7

1216

Checkpoint}RS is a radius of (R. Is }ST tangent to (R?

In the diagram, }RS is a radius of (R.R

S

T1024

26Is }ST tangent to (R?

SolutionUse the Converse of the Pythagorean Theorem. Because 102 1 242 5 262, nRST is a and }RS ⊥ . So, is perpendicular to a radius of (Rat its endpoint on (R. By , }ST is to (R.

Example 4 Verify a tangent to a circle

ga2nt-06.indd 120 4/16/07 9:01:30 AM



Your Notes

THEOREM 6.2

P

R

S

T

Tangent segments from a commonexternal point are .

}QR is tangent to (C at R R

S

C

Q3x 1 5

32and }QS is tangent to (Cat S. Find the value of x.

Solution QR 5 QS Tangent segments from a common

external point are .

5 Substitute.

5 x Solve for x.

Example 6 Use properties of tangents

7. In the diagram, K is a point of K

J

r

rL 56

32

tangency. Find the radius r of (L.

Checkpoint Complete the following exercise.

TRIANGLE SIMILARITY POSTULATES AND THEOREMS

Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are to two angles of another

, then the two triangles are .

Theorem 6.3 Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are , then the triangles are .

Theorem 6.4 Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is to an angle of a second triangle and the lengths of the sides including these angles are , then the triangles are .

ga2nt-06.indd 121 4/16/07 9:01:31 AM



Your Notes

8. }RS is tangent to (C at S and

RT

S

C

x2

49}RT is tangent to (C at T.Find the value(s) of x.

9. In the diagram, }ST is a common

PM

S

N

T

internal tangent to (M and (P.Use similar triangles to show that

MN}PN 5

SN}TN.


Homework

In the diagram, both circles are

B E

A

C D

centered at A. }BE is tangent to theinner circle at B and }CD is tangentto the outer circle at C. Use similartriangles to show that AB

}AC

5AE}AD

.

Solution}AB ⊥ }BE and }AC ⊥ }CD

Definition of ⊥

All right ? are ù.

∠CAD ù ∠BAE

AA Similarity Postulate

Corr. side lengths of ,ns are proportional.

Example 7 Use tangents with similar triangles

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Match the notation with the term that best describes it.

1. D A. Center

B

A

D

C

H

F

G

E

2. @##$ FH B. Chord

3. }

CD C. Diameter

4. }

AB D. Radius

5. C E. Point of tangency

6. }

AD F. Common external tangent

7. @##$ AB G. Common internal tangent

8. @##$ DE H. Secant

Use the diagram at the right.

9. What are the diameter and radius of (A? y

x1

1

BA

10. What are the diameter and radius of (B?

11. Describe the intersection of the two circles.

12. Describe all the common tangents of the two circles.

Use (P to draw the part of the circle described or to answer the question.

13. Draw a diameter }

AB .

P

14. Draw tangent line @##$ CB .

15. Draw chord } DB .

16. Draw a secant through point A.

17. What is the name of a radius of the circle?

LESSON

6.1 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON

6.1 Practice continued

Tell how many common tangents the circles have and draw them.

18. 19. 20.

Draw two circles with the given number of common tangents.

21. 3 22. 2 23. 1

In the diagram, } BC is a radius of (C. Determine whether } AB is tangent to (C. Explain your reasoning.

24.

C

BA

25.

C

BA

43

5

26.

C

B

A63

7

In the diagram, } AB is tangent to (C at point B. Find the radius r of (C.

27.

C

BA

4

2 rr

28.

C

BA

12

8r

r

29.

C

BA

15

9r

r



} JK is tangent to (L at K and } JM is tangent to (L at M. Find the value of x.

30.

L

K

M

J

22

x 31.

J

K

M

L

3x

x 1 12

32. J K

M

L2x 1 2

5x 2 4

33. Softball On a softball fi eld, home plate is 38 feet from the pitching circle. Home plate is about 45.3 feet from a point of tangency on the circle.

rr

38 ft

45.3 ft

a. How far is it from home plate to a point of tangency on the other side of the pitching circle?

b. What is the radius of the pitching circle?

34. In the diagram, }

QT is tangent to both circles

RS

P

T and } RT is tangent to both circles. Use similar

triangles to show that PS

} ST

5 QR

} RT .

LESSON


Name ——————————————————————— Date ————————————



Goal p Use angle measures to find arc measures.

6.2 Find Arc MeasuresGeorgiaPerformanceStandard(s)

MM2G3b, MM2G3d

Your Notes

MEASURING ARCS

The measure of a minor arc is the measure of its central angle. The expression mCAB is read as “the measure of arc AB.”

The measure of the entire circle is B

A

C

D

508

. The measure of a major arc is the difference between and the measure of the related minor arc. mCAB 5 508

The measure of a semicircle is . mCADB 5 3108

VOCABULARY

Central angle

Semicircle

Arc

Minor arc

Major arc

Measure of a minor arc

Measure of a major arc

Congruent circles

Congruent arcs

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Your Notes

Money You join a new bank and

A

B

C

D

Checking558

Savings1058

MoneyMarket

1408

Bonds608

divide your money several ways, as shown in the circle graph at the right. Find the indicated arc measures.

a. mCBD b. mCBCD

Solutiona. mCBD 5 mCBA 1 mCAD b. mCBCD 5 3608 2 mCBD

5 1 5 3608 2

5 5

Example 2 Find measures of arcs

ARC ADDITION POSTULATE

The measure of an arc formed

C

BA

by two adjacent arcs is the sum of the measures ofthe two arcs. mCABC 5 m 1 m

1. Find mCRTS in the diagram at the right.

P

T

S

R

458


Find the measure of each arc of (C, where

F

EC

D1178}DF is a diameter.

a. CDE b. CDFE c. CDEF

a. CDE is a arc, so mCDE 5 m∠ 5 .

b. CDFE is a arc, so mCDFE 5 2 5 .

c. }DF is a diameter, so CDEF is a , andmCDEF 5 .

Example 1 Find measures of arcs

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Your Notes

2. In Example 2, find (a) mCBCA and (b) mCABC .

3. In the diagram at the right, is

P S

R

CPQ > CSR ? Explain why or why not.


Tell whether the given arcs are congruent. Explain why or why not.

a. CBC and CDE b. CAB and CCD c. CFG and CHJ

B E

C D

758 758

D

C

B

A808 F

G

H

J

Solution

a. CBC CDE because they are in

and mCBC mCDE .

b. CAB and CCD have the same , but they are because they are arcs of circles that

are .

c. CFG CHJ because they are in

and mCFG mCHJ .

Example 3 Identify congruent arcs

Homework

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Name the arc shown in bold.

1.

CD

A

B

2.

CD

A

B

3.

P

S

R

} AB and } FE are diameters of (C. Determine whether

C

F D

E

A

B

the given arc is a minor arc, major arc, or semicircle.

4. C AE 5. C AEB

6. C FDE 7. C DFB

8. C FA 9. C BE

10. C BDA 11. C FB

In (O, } MQ and } NR are diameters. Find the

ON R

M

P

30�

70�

indicated measure.

12. m C MN 13. m C NQ

14. m C NQR 15. m C MRP

16. m C QR 17. m C MR

18. m C QMR 19. m C PQ

20. m C PRN 21. m C MQN

LESSON

6.2 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the indicated arc measure.

22. m C AB

A

B

C

D

23. m C ACB

A

B

C

D

24. m C CA

A

B

C

D

Use the information given about a central angle of a circle to fi nd the measure of its corresponding arc.

25. The central angle is a right angle.

26. The central angle is a diameter.

27. The central angle is complementary to a 308 angle.

28. The central angle is supplementary to a 588 angle.



Tell whether C AB ù C CD . Explain.

29.

A

B

C

D

30. A

B

758

C

D

758

31. A B

C

D

658

658

32.

B

A

1108

D

C

1358

1158

Keeping Time In the clock face shown at the right,

1

2

3

4567

8

9

1011

12the positions of the numbers determine congruent arcs along the circle.

33. What is the measure of the arc between any two consecutive numbers?

34. An arc is traced out by the end of the second hand as it moves from the 12 to the 4. Is this a minor arc or a major arc?

35. Starting at the 2, what number does the end of the second hand reach as it completes a semicircle?

36. When the second hand moves from the 8 to the 3, what is the measure of the arc?

37. When the second hand moves from halfway between the 1 and the 2 to 4 } 5 of the way

from the 1 to the 2, what is the measure of the arc?

38. The second hand moves from the 3 to the 7. What is the measure of the corresponding major arc?

LESSON


Name ——————————————————————— Date ————————————



6.3 Apply Properties of ChordsGoal p Use relationships of arcs and chords in a circle.Georgia


MM2G3a, MM2G3d

Your Notes

In the diagram, (A > (D, B

C

A

E

FD

1258}BC > }EF , and mCEF 5 1258.Find mCBC .

Solution

Because }BC and }EF are congruent in congruent

, the corresponding minor arcs CBC and CEF are .

So, mCBC 5 mCEF 5 .

Example 1 Use congruent chords to find an arc measure

THEOREM 6.6

If one chord is a perpendicular bisector S

T

R

Pof another chord, then the first chord is a diameter.

If }QS is a perpendicular bisector of }TR, then is a diameter of the circle.

THEOREM 6.7

If a diameter of a circle is perpendicular E

F

G

D

Hto a chord, then the diameter bisects the chord and its arc.

If }EG is a diameter and }EG ⊥ }DF , then }HD > }HFand > .

THEOREM 6.5 B C

A D

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are CAB > CCD if and

only if > .congruent.

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Your Notes

Journalism A journalist is writing aa story about three sculptures, arranged as shown at the right. Where should the journalist place a camera so that it is the same distance from each sculpture?

SolutionStep 1 Label the sculptures A, B, and C. Draw segments

}AB and }BC.

Step 2 Draw the of }ABand }BC. By , these bisectors are diameters of the circle containing A, B, and C.

Step 3 Find the point where these bisectors .This is the center of the circle containing A, B, and C, and so it is from each point.

Camera

A

B

C

Example 2 Use perpendicular bisectors

1. If mCTV 5 1218, find mCRS . S

V

T

R6

6

2. Find the measures of CCB , CBE , C

E

AB D

4x 8

(80 2 x)8

and CCE .


ga2nt-06.indd 127 4/16/07 9:01:40 AM



Your NotesTHEOREM 6.8

EF

GAC

B

DIn the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from }AB > }CD if and only

if 5 .the center.

3. In the diagram in Example 3, suppose AB 5 27 and EF 5 GF 5 7. Find CD.


In the diagram of (F, AB 5 CD 5 12.

E

F

GA

C

B

D

12

12

7x 2 8

3x

Find EF.

SolutionChords }AB and }CD are congruent, so by Theorem 6.8 they are from F. Therefore, EF 5 .

EF 5 Use Theorem 6.8.

3x 5 Substitute.

x 5 Solve for x.

So, EF 5 3x 5 3( ) 5 .

Example 3 Use Theorem 6.8

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Your Notes Example 4 Use chords with triangle similarity

4. In Example 4, suppose in (S, QN 5 26, NR 5 24, ST 5 SU, }QN ⊥ }MP, and ∠NRQ is a right angle. Show that nPTS , nNRQ.


Homework

In (S, SP 5 5, MP 5 8, ST 5 SU, 8N

S

PMU

T

R

5

}QN ⊥ }MP, and ∠NRQ is a right angle. Show that nPTS , nNRQ.

1. Determine the side lengths of nPTS. Diameter }QN is perpendicularto }MP, so by }QNbisects }MP. Therefore,

PT 51}2 5

1}2 ( ) 5 . }SP has a given length

of . Because }QN is perpendicular to }MP, ∠PTS

is a and TS 5 Ï}}

( )2 2 ( )2 5

Ï}}

2 5 . The side lengths of nPTS are SP 5 , PT 5 , and TS 5 .

2. Determine the side lengths of nNRQ. The radius }SPhas a length of , so the diameter QN 5 2( ) 5

2( ) 5 . By }NR ù }MP, soNR 5 MP 5 . Because ∠NRQ is a ,

RQ 5 Ï}}

( )2 2 ( )2 5 Ï}}

25 . The side lengths of nNRQ are QN 5 , NR5 , and RQ 5 .

3. Find the ratios of corresponding sides. PT}NR 5 5 ,

TS}RQ 5 5 , and SP

}QN 5 5 . Because the

side lengths are proportional, nPTS , nNRQ by the .

ga2nt-06.indd 129 4/16/07 9:01:42 AM



Name ——————————————————————— Date ————————————

LESSON

6.3 PracticeFind the chord length.

1. AB 2. FG 3. KM

1558

15589

B

A D

C

6

E F

G

35

K

J M

L

Find the arc length.

4. m C XY 5. m C TU 6. m C AB

1058

X

W

Z

Y

858

S

R U

T 1008

B

A

C

Tell whether } QS is a diameter of the circle. If not, explain why.

7.

P R

S

8.

P R

S

19

20

9.

T

R

S

99



Tell whether the measures are equal.

10. LK and KN 11. m C ST and m C RS 12. m C BC and m C CD

L K

M N

P

T

R

US

C

E

D

B

Find the given measure.

13. BD 14. m C JK 15. m C MN

C

E

D

B A5

H

K

J

G

L558

M

P

N

L 618

Tell whether the lengths are equal.

16. CD and EF 17. JK and LM 18. TQ and UQ

C

D

F

E

J

K

M

L9

8

N

P

S

R

UT9 9

888

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the given measure.

19. AQ 20. HJ 21. m C TU

GF

E

C

A

D18

18

7

LH

J K

4 4 7

UR

S T

958

Find the value of x.

22.

E

B

D

C

6x 1 1

8x 2 5

23.

A

B

CQ

4x 2 1 3x 1 6 24.

X

Y

Z

(35x 2 16)8

31x8

25. Coins You notice that the word LIBERTY on the heads side of a quarter makes about the same arc as the words QUARTER DOLLAR on the tails side. Explain how you can use a straight ruler to check whether this is true.

26. In (C, } FE ⊥ }

GB , ∠ FDE is a right angle, }

AC ù }

CH ,

C A

16

20

F

H

D

E

G

B

DE 5 16, and FE 5 20. Show that nCAB , nFDE.



6.4 Use Inscribed Angles and PolygonsGoal p Use inscribed angles of circles.Georgia


MM2G3b, MM2G3d

Your Notes

THEOREM 6.9: MEASURE OF AN A

CD

B

INSCRIBED ANGLE THEOREM

The measure of an inscribed angle is one half the measure of its intercepted arc.

m∠ADB 51}2

Find the indicated measure in (P. Q

PT

R

S

378608a. m∠S b. mCRQ

Solution

a. m∠S 51}2 5

1}2 ( ) 5

b. mCQS 5 2m∠ 5 2 p 5 .

Because CRQS is a semicircle, mCRQ 5 1808 2 5 1808 2 5 .

Example 1 Use inscribed angles

VOCABULARY

Inscribed angle

Intercepted arc

Inscribed polygon

Circumscribed circle

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Your Notes

1. m∠GHJ 2. mCCD

G

H

F

J1008

D

BC408

Checkpoint Find the indicated measure.

THEOREM 6.10 A

C

D

BIf two inscribed angles of a circle intercept the same arc, then the angles are congruent. ∠ADB > ∠

THEOREM 6.11

If a right triangle is inscribed in a

C

D

A

B

circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the

m∠ABC 5 908 if and only if is a diameter of the circle.

angle opposite the diameter is the right angle.

THEOREM 6.12

A quadrilateral can be inscribed in a

D

E F

GC

circle if and only if its opposite angles are supplementary.

D, E, F, and G lie on (C if and only if m∠D 1 m∠F 5 m∠E 1 m∠G 5 .

ga2nt-06.indd 131 4/16/07 9:01:45 AM



Your Notes

Security A security camera that rotates 908is placed in a location where the only thing viewed when rotating is the wall. You want to change the camera’s location. Where else can it be placed so that the wall is viewed in the same way?

SolutionFrom Theorem 6.11, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a of the circle. So, draw the circle that has the width of the wall as a . The wall fits perfectly with your camera’s 908 rotation from any point on the in front of the wall.

Example 2 Use a circumscribed circle

3. Find m∠RTS.

T

SQ

R

318

4. A right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle?


ga2nt-06.indd 132 4/16/07 9:01:47 AM



Your Notes

Find the value of each variable.

a.

P

Q

R

S

1008

x 8

y 8

888

b.

J L

K

M

8x 8 4x 8

3y 8

3y 8

Solutiona. PQRS is inscribed in a circle, so its opposite angles

are .

m∠P 1 m∠R 5 m∠Q 1 m∠S 5

1 y8 5 1 x8 5

y 5 x 5

b. JKLM is inscribed in a circle, so its opposite angles are .

1 m∠L 5 1 m∠M 5

1 4x8 5 1 3y8 5

5 5

x 5 y 5

Example 3 Use Theorem 6.12

5. Find the values of a and b.A

C

B

D

5a8

968

4b8


Homework

ga2nt-06.indd 133 4/16/07 9:01:48 AM



Find the indicated measure.

1. m∠A 2. m∠A 3. m∠B

A

B

C

1588

C24�

A

B

A

B

C

4. m C BC 5. m C BC 6. m C BC

A

B

C408

A

B

C

768

A

B C

7. m∠C 8. m∠A 9. m C BC

A

B

C

508

A

B

C328

A

BC

648

LESSON

6.4 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the indicated measure in (M.

Q

N

M

OP 688

1188

10. m∠PNO

11. m∠QNP

12. m C PQ

13. m C QO

14. m∠NMO

15. m C NOP

16. m∠QMP

17. m C OQN



Name two pairs of congruent angles.

18.

C

F

E

D

19.

Z

YX

W

Decide whether a circle can be circumscribed about the quadrilateral.

20.

130�

92�

70�

21. 22.

120�

Find the values of the variables.

23.

628 628

x 8 y 8

A

B C

D

24.

688

1048

x 8 y 8

O

M N

P

25.

988

x 8y 8

C

DE

F

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the values of the variables.

26.

408

1708

x 8

y 8

U

R S

T

27.

808528

1148

x 8

y 8

M

J

K

L

28. 758

1048

1088x 8

y 8

H

E

G

F

Find m∠A and m∠C.

29.

328

408

A

B

D

C 30.

738

318

A

B

D

C

31. 458

(2x 1 8)8

(3x 2 24)8A

B

D

C

32. Stained Glass You are making the stained glass ornament

A

B

D

C

shown at the right. The kite is symmetric, so ∠ A > ∠C, } BD is a diameter of the circle, and m∠D 5 608. What are the measures of ∠ A, ∠B, and ∠C?



6.5 Apply Other Angle Relationships in CirclesGoal p Find the measures of angles inside or outside

a circle.

GeorgiaPerformanceStandard(s)

MM2G3b, MM2G3d

Your Notes

THEOREM 6.13C

A

B

2 1If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

m∠1 51}2

m∠2 51}2

Line m is tangent to the circle. Find the indicated measure.

a. m∠1

A

B

1 m

1328

b. mCEFD

F

D

E

1108

m

Solution

a. m∠1 5 (1328) 5

b. mCEFD 5 (1108) 5

Example 1 Find angle and arc measures

1. m CACB

1208

A B

C


ga2nt-06.indd 134 4/16/07 9:01:49 AM



Your NotesTHEOREM 6.14: ANGLES INSIDE THE CIRCLE

THEOREM

If two chords intersect

B

D

AC

12

inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

m∠1 51}2

(m 1 m )

m∠ 2 51}2

(m 1 m )

Find the value of x.F J

G H

1128

1408

x 8Solution

The chords }FH and }GJ intersect inside the circle.

x8 51}2

(m 1 m ) Use Theorem 6.14.

x8 51}2( 1 ) Substitute.

x 5 Simplify.

Example 2 Find an angle measure inside a circle

2.

528

1448

x 8FJ

HG

Checkpoint Find the value of x.

ga2nt-06.indd 135 4/16/07 9:01:51 AM



Your NotesTHEOREM 6.15: ANGLES OUTSIDE THE

CIRCLE THEOREM

If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

AB

C

1

P

Q

R

2

m∠1 51}2

(mCBC 2 mCAC ) m∠ 2 51}2

(mCPQR 2 mCPR )

W

X

Z

Y

3 m∠3 51}2

(mCXY 2 mCWZ )


H

J

F

G x 8 7281708Solution

The tangent ###$GF and the secant###$GJ intersect outside the circle.

m∠FGH 51}2

(m 2 m ) Use Theorem 6.15.

x8 51}2( 2 ) Substitute.

x 5 Simplify.

Example 3 Find an angle measure outside a circle

3. 268

308

y 8

F

J

K

H

G

Checkpoint Find the value of y.

Homework

ga2nt-06.indd 136 4/16/07 9:01:52 AM



Name ——————————————————————— Date ————————————

LESSON

6.5 PracticeFind the measure of each numbered angle or arc.

1.

1

172� 2.

12128�

3. 1

2

117�

4.

1

282�

5.

12

131�33�

6.

12 97�115�

7.

1 247� 41�

8.

1

2

132�

86�117�

9.

1

2

126�

70�66�

10. 1

2

270�

11.

143�92�

12.

1 80�32�



Find the measure of each numbered angle or arc.

13. 12

134�

170�

14.

1138�

66�

15.

12

105�

130�

70�


16.

38�

180�

x �

17. 105�

115�

x �

18. 96�

x �

19.

(3x 2 1)�

(5x 1 33)�

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON



20.

38�

(10x 2 1)�

x � 21. 45�

x �

22. Satellites A satellite is taking pictures of Earth from

8000 mi

4000 mi

U

S

R T

Not drawn to scale

4000 miles above its surface. What is the measure of C RT , which corresponds to the portion of Earth that can

be photographed from the satellite?

23. Theater A play is being presented on a circular stage. B

A

C D E

388

808The two main characters are at positions A and B at the back of the stage. Use the diagram to answer the following questions.

a. What angle of view between the main characters does an actor at position C at center stage have?

b. What angle of view between the main characters does the orchestra conductor at point D have?

c. What angle of view between the main characters does an audience member at point E have?



6.6 Find Segment Lengths in CirclesGoal p Find segment lengths in circles.Georgia


MM2G3a, MM2G3d

Your Notes

VOCABULARY

Segments of a chord

Secant segment

External segment

THEOREM 6.16: SEGMENTS OF CHORDS THEOREM

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of

EA

D

C

B

the other chord. EA p 5 EC p

Find ML and JK.

N

JM

L

K

x 1 1

x 1 5

x 1 3

x

NK p NJ 5 p

x p (x 1 5) 5 ( ) p ( )

x2 1 5x 5

x 5

Find ML and JK by substitution.

ML 5 ( ) 1 ( ) JK 5 1 ( )

5 1 1 1 5 1 1

5 5

Example 1 Find lengths using Theorem 6.16

ga2nt-06.indd 137 4/16/07 9:01:54 AM



Your Notes

1.

6

125

x

2.

7

8

6x

Checkpoint Find the value of x.


T

P

S

Q

R

46

5x

Solution

RQ p RP 5 RS p RT Use Theorem .

p ( 1 ) 5 p (x 1 ) Substitute.

5 x 1 Simplify.

5 x Solve for x.


THEOREM 6.17: SEGMENTS OF SECANTS THEOREM

If two secant segments share the same endpoint outside a circle, then the product of the lengthsof one secant segment and its external segment equals the product of the lengths of the other secant segment and its

E

B

C

AD

external segment. p EA 5 ED p

ga2nt-06.indd 138 4/16/07 9:01:55 AM



Your Notes

3. Find JK.

J

K

L

M 8

6

x


Find RS.. R Q

S

T

14

12

x

Solution

RQ2 5 RS p Use Theorem .

2 5 x p ( ) Substitute.

5 x2 1 x Simplify.

0 5 x2 1 x 2 Write in standard form.

x 56 Ï

}}}2 2 4( )( )

}}}2( )

Use quadratic formula.

x 5 Simplify.

Lengths cannot be , so use the solution.

So, x 5 < , and RS < .


THEOREM 6.18: SEGMENTS OF SECANTS AND TANGENTS THEOREM

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the

E

A

C

D

tangent segment. EA2 5 p

Homework

ga2nt-06.indd 139 4/16/07 9:01:57 AM



Name ——————————————————————— Date ————————————

LESSON

6.6 PracticeFill in the blanks. Then fi nd the value of x.

1. x p _____ 5 5 p _____ 2. 6 p _____ 5 3 p _____ 3. x p _____ 5 8 p _____

3

5

9

x

3

4

6

x

48

6

x

4. 4 p _____ 5 5 p _____ 5. 3 p _____ 5 4 p _____ 6. 3 p _____ 5 5 p _____

13

5

4

x

3

48

x

3

5

10

x

7. x2 5 4 p _____ 8. x2 5 2 p _____ 9. x2 5 6 p _____

4

12 x

2

16

x

4

6

x




10. 5

6

10x

11.

6

8

3x

12. 2

5

x

13.

7

3

4x

14.

5

64

x

15.

10

8

8

x

16.

3

24

x

17.

12

16

x

18. 8

x

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON



19. 7

6

4

x

20. 3

18

2x

3x

21.

31

20

x

22.

3

6

9

x

23.

139

5x

24.

8

x

3x

x � 2

25. Doorway An arch over a doorway is an arc that

E

B

O

A50 cm

160 cm

C D

is 160 centimeters wide and 50 centimeters high. You are curious about the size of the entire circle containing the arch. By drawing and labeling a circle passing through the arc as shown in the diagram, you can use the following steps to fi nd the radius of the circle.

a. Find AB.

b. Find AC and AD.

c. Use AB, AC, and AD to fi nd EA.

d. Find EB.

e. Find the length of the radius, EO.



6.7 Circumference and Arc LengthGoal p Find arc lengths and other measures of circles.Georgia


MM2G3c

Your Notes

THEOREM 6.19: CIRCUMFERENCE OF A CIRCLE

The circumference C of a circle is

dC

rC 5 or C 5 , where dis the diameter of the circle and r is the radius of the circle. C 5 5

VOCABULARY

Circumference

Arc length


a. Circumference of a circle b. Radius of a circle with with radius 11 meters circumference 18 yards

Solutiona. C 5 2πr b. C 5 2πr

5 2 p π p 5 2πr

5 π 5 r

< m yd < r

Example 1 Use the formula for circumference

1. Find the circumference of a circle with diameter 23 inches.


ga2nt-06.indd 140 4/16/07 9:01:58 AM



Your NotesARC LENGTH COROLLARY

In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 3608.

Arc length of CAB}}

2πr 5mCAB}3608

, or r

A

BP

Arc length of CAB 5mCAB}3608

p 2πr


a. Arc length of CAB b. mCRS

2 m888

A

B

P12.3 ft

38 ft

S

R

a. Arc length of CAB 5 p 2π( ) < meters

b.Arc length of CRS}}

2πr 5mCRS}3608

Write equation.

2π1 25

mCRS}3608

Substitute.

p2π1 2

5 mCRS Multiply each side by .

< mCRS Use a calculator.

Example 2 Find and use arc lengths

2. Arc length of CAB 3. Circumference of (Z

4 ft

978

A

B

P 3.44 cm608

X

Y

Z


Homework

ga2nt-06.indd 141 4/16/07 9:01:59 AM



Use the diagram to fi nd the indicated measure.

1. Find the circumference. 2. Find the circumference. 3. Find the radius.


4. The exact radius of a circle with circumference 36 meters

5. The exact diameter of a circle with circumference 29 feet

6. The exact circumference of a circle with diameter 26 inches

7. The exact circumference of a circle with radius 15 centimeters

Find the length of C AB .

8. 9. 10.

LESSON

6.7 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


In (D shown below, ∠ ADC > ∠ BDC. Find the indicated measure.

11. m C ACB 12. m C CB

13. Length of C ACB 14. Length of C CB

15. m C ABC 16. Length of C BAC


17. Length of C AB 18. Circumference of (Q 19. Radius of (Q



Find the perimeter of the region.

20. 21.

22. In the table below, C AB refers to the arc of a circle. Complete the table.

Radius 3 9 12 10.5

m C AB 608 358 1458 2908

Length of C AB 17.28 5.19 12.65 16.76

23. Scooter Wheel The wheel on a manual scooter has a

4 in.12

diameter of 4 1 }

2 inches, as shown.

a. Find the circumference of the wheel.

b. How many feet does the wheel travel when it makes 150 revolutions? Round to the nearest foot.

24. Birthday Cake A birthday cake is sliced into 8 equal pieces. The arc length of one piece of cake is 6.28 inches, as shown. Find the diameter of the cake.

LESSON


Name ——————————————————————— Date ————————————



6.8 Areas of Circles and SectorsGoal p Find the areas of circles and sectors.Georgia


MM2G3c, MM2G3d

Your Notes

VOCABULARY

Sector of a circle

THEOREM 6.20: AREA OF A CIRCLEr

The area of a circle is π times the square of the radius.

A 5


a. Area4.2 m

b. Diameter

A 5 201 in.2

Solutiona. A 5 πr2 Write formula for area of a circle.

5 π( )2 Substitute for r.

5 π Simplify.

< m2 Use a calculator.

b. A 5 πr2 Write formula for area of a circle.

5 πr2 Substitute for A.

5 r2 Divide each side by .

in. < r Find the positive square root of each side.

The radius is about inches, so the diameter is about inches.

Example 1 Use the formula for area of a circle

ga2nt-06.indd 142 4/16/07 9:02:01 AM



Your NotesTHEOREM 6.21: AREA OF A SECTOR

The ratio of the area of a sector of a circle to the area of the whole circle (πr2) is equal to the ratio of the measure of the intercepted arc to 3608.

Area of sector APB5

3608, or

r

P

A

B

Area of sector APB 53608

p

Find the areas of the sectors formed

5638

R

PS

by ∠RQS.

SolutionStep 1 Find the measures of the minor and major arcs.

Because m∠RQS 5 , mCRS 5 and

mCRPS 5 3608 2 5 .

Step 2 Find the areas of the small and large sectors.

Area of small sector 5 mCRS}3608

p πr2

53608

p π p 2

<

Area of large sector 5 mCRPS}3608

p πr2

53608

p π p 2

<

The areas of the small and large sectors are about square units and square units,

respectively.

Example 2 Find areas of sectors

ga2nt-06.indd 143 4/16/07 9:02:02 AM



Your Notes

Use the diagram to find the area

458

A

A 5 22 m2 C

B

of (C.

Solution

Area of sector ACB 5mCAB}3608

p Area of (C

53608

p Area of (C

5 Area of (C

The area of (C is square meters.

Example 3 Use the Area of a Sector Theorem

1. Find the radius of the circle.

A 5 154 ft2

2. Find the areas of the sectors

1068

9 ft

P R

S

formed by ∠RQS.

3. Find the area of (G.

808

F

A 5 6.4 yd2G

H


Homework

ga2nt-06.indd 144 4/16/07 9:02:03 AM



Find the exact area of the circle. Then fi nd the area of the circle to the nearest hundredth.

1. 2. 3.


4. The area of a circle is 58 square inches. Find the radius.

5. The area of a circle is 37 square meters. Find the radius.

6. The area of a circle is 106 square centimeters. Find the diameter.

7. The area of a circle is 249 square feet. Find the diameter.

Find the areas of the sectors formed by ∠ ACB.

8. 9. 10.

LESSON

6.8 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Use the diagram to fi nd the indicated measure.

11. Find the area of (S. 12. Find the area of (S. 13. Find the radius of (S.

The area of (Z is 124.44 square centimeters. The area of sector XZY is 28 square centimeters. Find the indicated measure.

14. Radius of (Z 15. Circumference of (Z

16. m C XY 17. Length of C XY

18. Perimeter of shaded region 19. Perimeter of unshaded region



Find the area of the shaded region.

20. 21. 22.

23. Hockey A face-off circle from a hockey rink is shown at the right. The diameter of the circle is 30 inches. Find the area of the face-off circle.

24. Pizza A pizza is cut into 8 congruent pieces, as shown. The diameter of the pizza is 16 inches. Find the area of one piece of pizza.

25. Clock A wall clock has an area of 452.39 square inches. Find the diameter of the clock. Then fi nd the area of the sector formed when the time is 3:00, as shown.

LESSON


Name ——————————————————————— Date ————————————



6.9 Surface Area and Volume of SpheresGoal p Find surface areas and volumes of spheres.Georgia


MM2G4a, MM2G4b

Your Notes

THEOREM 6.22: SURFACE AREA OF A SPHERE

The surface area S of a sphere is rS 5 , where r is the radius

of the sphere.

VOCABULARY

Sphere

Center of a sphere

Radius of a sphere

Chord of a sphere

Diameter of a sphere

Great circle

Hemisphere

ga2nt-06.indd 145 4/16/07 9:02:05 AM



Your Notes

Find the surface area of the sphere.

7 cm

Solution

S 5 4πr2 Formula for surface area of a sphere

5 4π( )2 Substitute for r.

5 π Simplify.

< Use a calculator.

The surface area of the sphere is about square centimeters.

Example 1 Find the surface area of a sphere

The circumference of a sphere is 12π feet. Find the surface area of the sphere.

Solution C 5 2πr Formula for circumference

5 2πr Substitute for C.

5 r Divide each side by .

S 5 4πr2 Formula for surface area

5 4π( )2 Substitute for r.

ø Use a calculator.

The surface area of the sphere is about square feet.

Example 2 Find the circumference of a sphere

ga2nt-06.indd 146 4/16/07 9:02:06 AM



Your Notes

1. The diameter of a sphere is 1}Ï

}

π meter. Find the

surface area of the sphere.

2. The circumference of a sphere is 2π inches. Find the surface area of the sphere.


THEOREM 6.23: VOLUME OF A SPHERE

The volume V of a sphere is r

V 5 , where r is the radius

of the sphere.

Find the volume of the sphere.

3 mm

Solution

V 54}3πr3 Formula for volume of a sphere

54}3π( )3 Substitute for r.

< Use a calculator.

The volume of the sphere is about cubic millimeters.

Example 3 Find the volume of a sphere

ga2nt-06.indd 147 4/16/07 9:02:07 AM



Your Notes

A sphere has a radius of 6 meters. The radius of a second sphere is 3 meters, which is one half the radius of the first sphere. How do the surface area and volume of the second sphere compare to the surface area and volume of the first sphere?

Solution

First sphere: Substitute for r and simplify.

S 5 4πr2 5 4π( )2 5 π m2

V 5 4}3πr3 5

4}3π( )3 5 π m3

Second sphere: Substitute for r and simplify.

S 5 4πr2 5 4π( )2 5 π m2

V 5 4}3πr3 5

4}3π( )3 5 π m3

The surface area of the second sphere is , or

the surface area of the first sphere. The volume of the

second sphere is , or the volume of the

first sphere.

Example 4 Determine the effects of a change in radius

3. The radius of a sphere is 2.4 centimeters. Find the volume of the sphere. Round your answer to two decimal places.

4. The radius of a sphere is 4 inches. Explain how the surface area and volume change if the radius is doubled.


Homework

ga2nt-06.indd 148 4/16/07 9:02:08 AM



Name ——————————————————————— Date ————————————

LESSON

6.9 Practice 1. Name the center of the sphere.

P S

R

T

7 m

2. Name a chord of the sphere.

3. Name a radius of the sphere.

4. Name a diameter of the sphere.

5. Find the circumference of the great circle that contains P and S. Write your answer in terms of π.

Find the surface area of the sphere. Round your answer to two decimal places.

6.

6 cm

7.

19 ft

8.

22 in.

9.

2.5 m

10.

15 yd

11.

30 cm



In Exercises 12–15, use the diagram. The center of the sphere is C and its circumference is 17π feet.

12. What is half of the sphere called?

C

13. Find the radius of the sphere.

14. Find the diameter of the sphere.

15. Find the surface area of half of the sphere.

Find the radius of a sphere with the given surface area S.

16. S 5 324π cm2 17. S 5 4π ft2 18. S 5 163.84π m2

19. The circumference of a sphere is 338π meters. What is the surface area of the sphere? Round your answer to two decimal places.

Find the volume of the sphere. Round your answer to two decimal places.

20.

5 m

21.

14 in.

22.

6 ft

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the volume of the sphere. Round your answer to two decimal places.

23.

4.5 cm

24.

21 yd

25.

28 cm

Find the radius of a sphere with the given volume V.

26. V 5 2304π yd3 27. V 5 36π in.3 28. V 5 33.51 mm3

29. A sphere is inscribed in a cube of volume 8 cubic meters. What are the surface area and volume of the sphere? Round your answers to two decimal places.

In Exercises 30–32, use the following information.

Beach Ball A beach ball has a surface area of about 78.54 square feet.

S 5 78.54 ft2

30. Find the radius of the beach ball.

31. Find the circumference of a great circle of the beach ball. Round your answer to two decimal places.

32. Find the volume of the beach ball. Round your answer to two decimal places.

33. Planets The mean radius of Earth is approximately Earth

Mars

6378 km 3397 km

6378 kilometers. The mean radius of Mars is

approximately 3397 kilometers, or about 1 }

2 the

mean radius of Earth. How does the surface area of Mars compare to the surface area of Earth?



Words to ReviewGive an example of the vocabulary word.

Circle

Chord

Tangent

Arc, minor arc, major arc

Measure of a minor arc

Center, radius, diameter

Secant

Central angle

Semicircle

Measure of a major arc

ga2nt-06.indd 149 4/16/07 9:02:09 AM



Congruent circles

Inscribed angle

Inscribed polygon

Segments of a chord

External segment

Congruent arcs

Intercepted arc

Circumscribed circle

Secant segment

Circumference

ga2nt-06.indd 150 4/16/07 9:02:11 AM



Arc length

Sphere, center, radius, chord, diameter

Hemisphere

Sector of a circle

Great circle

ga2nt-06.indd 151 4/16/07 9:02:13 AM


Documents

Use Properties of Tangents - Ms. Young · Use Properties of Tangents Georgia Goal pUse properties of a tangent to a circle. Performance Standard(s) MM2G3a, MM2G3d Your Notes VOCABULARY